Monday, 29 August 2011

Okay, so you probably know what an Erdos number is - the number of links - where a link is a co-published paper - between Paul Erdos and yourself. You also probably know that Erdos numbers are used as a tongue-in-cheek measure of mathematical status. Much more irrational, of course, is the fact that logicians and others in more technical areas of philosophy tend to brag about their Erdos numbers (mine is 7, by the way - Erdos-Shelah-Woodin-Welch-Rayo-Uzquiano-Shapiro-Cook).

What is much more interesting than closeness to Erdos, however, is what all of this implies about the connectivity with respect to collaboration of the mathematical community. For example, as of the year 2000, approximately 98.4 percent of authors listed on Math Reviews who have an Erdos number at all have an Erdos number no higher than 7 (99.6% are no higher than 8, and 79.1% are no higher than 5. About 2/3 of authors published in Math reviews have an Erdos number).

Erdos number distribution:

504 have Erdos Number 1.
6593 have Erdos Number 2.
33605 have Erdos Number 3.
83642 have Erdos Number 4.

Setting Erdos aside, the average distance between any two mathematicians (again, where they are connected at all) is 7.64.

This got me thinking about philosophy, and how connected it is via collaboration in comparison with mathematics (and other disciplines). So I picked a random person (myself) and computed Cook numbers. Here are the results:

4 have Cook Number 1.
19 have Cook Number 2.
39 have Cook Number 3.
~250 have Cook Number 4.

Unlike the Erdos numbers, however, these numbers don't look like they will taper off anytime soon. Instead the numbers seem likely to continue increasing. For example, there are a number of philosophers who have collaborated a good bit, including Barry Smith, Walter Sinnott-Armstrong, and Otavio Beuno, who have Cook number 4.

Part of the issue is that I started with myself, and I have few collaborators (although my recent work in aesthetics did help the list become rather varied speciality-wise rather quickly).

Anyway, Cook numbers are merely suggestive (and silly as well). Nevertheless, the numbers do seem to grow steadily. It would be interesting to know what the collaboration graph for philosophy looks like in comparison to mathematics. Some interesting questions (which I would love to hear people's opinions/guesses on):

(1) What is the average length of the chain between two philosophers (who are connected at all)?

I am going to hazard a guess that the average chain length is similar to the length in math. My guess would be that the difference will be the number of distinct chains between two people. Keep in mind that some of the slower growth suggested by the numbers above will be offset by the smaller population of philosophers as compared to mathematicians.

(2) Within philosophy, is there a single, large 'network' as there is in mathematics?

Probably (see below). For example, as noted above, 2/3 of all authors listed on Math Reviews are in the network connected to Erdos, including a surprising number of people who strictly speaking aren't mathematicians, such as myself. Interestingly, the average number of collaborators for people in the large network is 4.73, while the average for people who aren't (but who have collaborated) is 1.63.

(3) What is the average number of collaborators within philosophy?

For example, on average an author listed in Math Reviews has 3.36.

(4) Is there any way to get access to the Phil Papers data to actually find any of this out?

Anyway, here is a challenge, just to make it interesting: I suspect that the answer to question (2) is "yes", and that the 'big' network includes the majority of philosophers in American and UK philosophy departments (and, obviously, I think that the network is the one I mapped a bit of above). Further, I make the following conjecture:

Any American or UK philosopher who has collaborated with at least four distinct people has a Cook number.

(Note that this is equivalent to the claim that any philosopher who has collaborated with four other people has an Erdos number)

Prove me wrong?

Bonus Question: Do any philosophers have Erdos-Bacon numbers (the sum of one's Erdos number and one's Bacon number, computed in terms of distance from Kevin Bacon via co-starring in a film). I know of one.

This actually raises a pretty interesting question (for someone like me interested in both logic and in comics). What sorts of content can be represented purely pictorially (i.e. in cartoons that only use (non-textual) pictures and conventional constructions like word balloons and the like (but without text in the balloons). Are there analogues of the semantic paradoxes that can be constructed non-linguistically in this way?

Monday, 22 August 2011

I am now working on the chapter of my book (on formal languages) which focuses on the historical development of formal languages and mathematical formalisms more generally. I wish I could go into these developments much more thoroughly than I will be able to in the book (given word limit and time constraints), as they are truly fascinating. (It is no secret to anyone having read my previous posts that I am obsessed with the topic of notations and symbolic systems!) So for now, let me share some bibliographical suggestions and a digest of my main findings.

Some of you may be thinking that this topic is not of obvious interest in the context of a blog on mathematical philosophy. But one of the hallmarks of mathematical philosophy is certainly the use of mathematical formalisms and special notational systems, and thus a better understanding of this methodology (including its history) does seem to fall within the remit of the blog. (At any rate, feel free to stop reading if you don't care much about history!) For those who are still with me, here we go.

A question that had puzzled me for years is: where does the explosive progress in mathematical notation of the 16th and 17th centuries come from? I knew enough about Latin medieval *academic* mathematics to know that nothing in the quadrivium curriculum seemed to anticipate the birth of modern algebra with Viete and Descartes. So it had to come from somewhere else, but where? Well, last year I attended a conference in Nancy, and the mystery stated to be dispelled with a talk by Albrecht Heeffer on the medieval abbaco tradition. Here is a passage from the paper corresponding to that talk:

By the end of the fifteenth century there existed two independent traditions of mathematical practice. On the one hand there was the Latin tradition as taught at the early universities and monastery schools in the quadrivium. Of these four disciplines arithmetic was the dominant one withDe Institutione Arithmeticaof Boethius as the authoritative text. Arithmetic developed into a theory of proportions as a kind of qualitative arithmetic rather than being of any practical use, which appealed to esthetic and intellectual aspirations. On the other hand, the south ofEuropealso knew a flourishing tradition of what Jens Hoyrup (1994) calls “sub-scientific mathematical practice”. Sons of merchants and artisans, including well-known names such as Dante Alighieri and Leonardo da Vinci, were taught the basics of reckoning and arithmetic in the so-called abbaco schools in the cities of North Italy, theProvence, andCatalonia. The teachers ormaestri d’abbacoproduced between 1300 and 1500 about 250 extant treatises on arithmetic, algebra, practical geometry and business problems in the vernacular. The mathematical practice of these abbaco schools had clear practical use and supported the growing commercialization of European cities. These two traditions, with their own methodological and epistemic principles, existed completely separately. (Heeffer, forthcoming)

Basically, the abbaco tradition was the missing link between Arabic algebra as consolidated inal-Khwārizmī’s “Book on restoration and opposition” (which in turn was inspired by other mathematical traditions, such as the Indian tradition) and the algebra of Viete and Descartes. The interesting thing is that Viete himself emphasizes his indebtedness to three Greek mathematicians (Pappus, Diophantus and Eudoxus -- seethis paper by Danielle Macbeth), but makes no reference to either Arabic algebra or to the sub-scientific abbaco schools (this fits well the Renaissance ethos of the time, going back to the Classics!). But besides the 'canonical' Arabic tradition, the abbaco tradition was much inspired by the introduction of special symbols and techniques to operate with the symbolism emerging in the mathematical tradition of the Maghreb in the later Middle Ages (again, it makes sense, as the abbaco people were merchants and thus traveled a lot!). The Maghrebian tradition appears to be the historical place of birth for many of the notational conventions still widely used, such as the notation for fractions.

Thus, one lesson to be learned here is that focusing only on the official, 'academic' story is simply not enough to understand the emergence of modern mathematical symbolism; the sub-scientific tradition of the abbaco schools is a crucial piece in the puzzle.

- Anotherpaper by Hoyrup, a concise survey of proto-algebra and pre-modern algebra.

- Chapter 5 of Bellos'Alex's Adventures in Numberland, which I also mentioned in my previous post. In it (p.181), we discover for example that the reason why we usexas the main symbol for the unknown is because it is one of the least used letters in French! Descartes introduced the convention that letters towards the end of the alphabet would be used for unknown quantities (while those at the beginning would be used for known quantities), but as hisLa Geometriewas being printed, the printer was running out of letters (think of the old-fashioned printing method of using small lead letters to imprint the paper). He asked if it mattered whetherx,yorzwas used; since it did not matter, he opted forxsimply because it is used less frequently in French. And here we are, still stuck withx's!

UPDATE: Kai von Fintel makes the excellent suggestion of adding some links to the (still) definitive account of the history of mathematical notations: Florian Cajori's A History of Mathematical Notations(1928).The first volume can be downloaded (for free) here, the second seems not to have been scanned yet. And here is a compilation (from Cajori's book) of earliest uses of various mathematical notations.

Sunday, 21 August 2011

As is well-known, the Leibnizian definition of identity, using the defining formula $\forall X(Xx \rightarrow Xy)$, is second-order. It quantifies over properties. One can get further information about this kind of definition of identity by seeing how it can be proved in a theory in which identity is already definable. In fact, one can consider any formula which behaves "like identity" in a given (second-order) theory, and then show that this formula is provably equivalent to both $\forall X(Xx \rightarrow Xy)$ and to $\forall R(\forall zRzz \rightarrow Rxy)$.

Suppose that $L$ is a second-order language, where we do not assume $=$ is a primitive. Let $T$ be an $L$-theory containing Comprehension for all arities (we just need unary and binary below).

Definition: An $L$-formula $x \approx y$ is a Leibniz formula in $T$ just in case both the following hold:

Friday, 19 August 2011

The summer break is officially over for me since Monday, but it's been difficult to find time to resume blogging here at M-Phi; I have lots of ideas for posts, but other pressing and time-consuming tasks keep interfering with my plans!

But anyway, here's a post at long last. So, one of the things I did during the summer break was to read Alex Bellos' Alex's Adventures in Numberland. (I'm your typical happy nerd: I read popular science books during the holidays...) It's a highly enjoyable book, which I am sure would interest even people who know a lot more math than the author himself. What I like about Bellos' books (as in his previous book on Brazilian football, Futebol) is that he brings in 'the human factor' in a empathetic, funny and insightful way, so the book is about math in all its different manifestations in people's lives. It ranges from high-level professional math to recreational math, going through gambling and numerology. I hope to be able to write a more extensive review of the book for M-Phi soon. (Ok, I'm not exactly neutral: a guy who loves math, loves Brazil and writes well, what's there not to like about him?)

One of the things I learned in the book is that the most successful technique so far for representing the hyperbolic plane is with crochet; yes, you heard me (well, read me), crochet. Here's a first sample:

The history of the discovery of hyperbolic planes is well known: in the 19th century, non-Euclidean geometry was discovered independently by Bolyai and Lobachevsky, and entailed (among many other things) that there are three kinds of planes/surfaces, depending on the number of lines parallel to a given line passing through a point not in that line: flat planes (which are described by Euclidean geometry -- exactly one line); spherical planes (no line); and hyperbolic planes (infinite number of lines).

Representing/visualizing flat and spherical planes was a piece of cake; flat planes was what geometers had been talking about for thousands of years, and spherical planes define for example the 'geometry' of the Earth (latitudes and longitudes). But "the challenge of visualizing the hyperbolic plane galvanized many mathematicians in the final decades of the nineteenth century." (Bellos 2010, 392) Poincare' had a good try with his disc model, which later provided inspiration for some beautiful woodcuts by Escher:

While clever, Poincare's disk model was still not sufficiently realistic. Mathematicians almost despaired of ever finding more realistic models when Hilbert proved in 1901 that it is impossible to describe a hyperbolic surface using a formula (this Hilbert guy, he just keeps popping up everywhere!). In practice, this also means that computers cannot create images of hyperbolic surfaces in a straightforward way (but I can imagine that, with some 'cheating', one could get close enough).

It was only in the 1970s that systematic attempts at representing hyperbolic planes were made again, and this time on a much more low-tech level, by Fields medalist Thurston. He construed paper models of hyperbolic planes, sticking together horseshoe-shaped slivers of paper. But the models would fall apart as soon as one would manipulate them, which didn't really help in terms of giving people a 'feel' for what hyperbolic planes look like.

But Thurston's attempt inspired mathematician Daina Taimina to think of a different, less fragile material to be used for representing hyperbolic planes. She first experimented with knitting, but it turned out not to be very appropriate for the enterprise (don't ask me, knitting is like rocket-science to me), so she switched to crochet. And thus were born the first crochet representations of hyperbolic planes! (That was in 1997.) The basic idea is the following: "start with a line of stitches, and then for each subsequent line add a fixed amount relative to the number of stitches on the line before. [...] This, she [Taimina] hoped, would create a piece of fabric that became wider and wider -- as if expanding out from itself hyperbolically." (Bellos 2010, 395)

Hyperbolic crocheting is now a important trend in crocheting circles, and perhaps slightly closer to the interests of M-Phi readers, it has had significant implications for the study of the geometry of hyperbolic planes. If nothing else, it gives mathematicians and students the chance to 'experience' something they only conceived of in purely theoretical, 'formalistic' ways: "Oh, so this is what a horocycle looks like!" But one can also do serious math in connection with crochet models of hyberbolic planes, as explained in a paper co-authored by Taimina and her husband David W. Henderson (both professors of mathematics at Cornell).

There are so many things I like about the idea of representing the hyperbolic plane with crochet. First, I am of course delighted that a craft traditionally associated with the female half of the humanity is used to make such an important contribution to mathematics. Second, it is a very nice case study of the interplay between 'doing the math' and visualizing a given concept such as that of hyperbolic planes, which is in first instance so far removed from what we can 'normally' represent. And finally, the crochet models are simply beautiful things! They look like corals, which is not surprising given the tendency in living beings to develop in a 'hyperbolic' way when they need to maximize their surface of contact with the external world. Just do a google images search with 'hyperbolic crochet' to see many more beautiful samples.

Sunday, 14 August 2011

Quine introduced the notion of weak discernibility in a 1976 Journal of Philosophy paper, "Grades of Discriminability". The criterion of weak discernibility is this. Suppose $L$ is an interpreted language. So, it makes sense to talk of the referents of syntactic strings in $L$. (The denotation of a term $t$; the extension of a predicate $P$, the range of the variables/quantifiers, the truth function associated any sentence connective $\frak{c}$.)
Suppose $a, b$ are in the range of $L$'s quantifiers. Then:

D1: $a$ is weakly discernible from $b$ in $L$ iff there is an irreflexive binary relation $R$ definable in $L$ such that $Rab$.

This looks, at first sight, very unfamiliar. Can we give an explanation of how it is related to more obvious analyses of identity?

First, let's negate both sides to get something like an indiscernibility notion. (Write "$x$ is strongly indiscernible from $y$ in $L$" to mean "$x$ is not weakly discernible from $y$ in $L$".) By logical manipulation [see below], we get:

D2: $a$ is strongly indiscernible from $b$ in $L$ iff for any reflexive binary relation $R$ definable in $L$, $Rab$.

Note that this quantifies over all (binary) relations definable in $L$. Let $Def_2(L)$ be the class of binary relations definable in $L$. Then Definition D2 is equivalent to:

So, the relation of strong indiscerniblity for $L$ is the smallest reflexive relation definable in $L$.
Now go back to how one might give a second-order definition of identity. The standard one, mentioned in the previous post is:

D5: $x = y$ iff $\forall X(Xx \rightarrow Xy)$.

But other definitions are possible. The identiy relation on a domain $D$ is the diagonal $\{(x,x): x\in D\}$. Clearly this is a reflexive relation. But notice that, in addition, it is the smallest reflexive relation on $D$.

D6: The identity relation on $D$ is the smallest reflexive binary relation on $D$.

With second-order logic, we can express this as follows:

D7: $x = y$ iff $\forall R(\forall xRxx \rightarrow Rxy)$.

Notice the similarity between this second-order definition of $=$ and the one using Quine's notion of "weak discernibility". The sole difference is really a question of definability. In fact, we can just rewrite Definition D4 as,

Notice the similarity of Definitions D7 and D8, which can be summarized:

Identity is the smallest reflexive relation.
Strong indiscernibility in $L$ is the smallest reflexive relation definable in $L$.

Note 1. The "logical manipulation" mentioned above requires two lemmas.
(1), a relation $R$ is reflexive iff its complement $\tilde{R}$ is irreflexive.
(2), a relation $R$ is definable in $L$ iff its complement $\tilde{R}$ is definable in $L$.

Note 2. One can re-express all this model-theoretically (see the papers Ketland 2006 and Ketland 2011 mentioned in the previous post.)
So, suppose we are given a language $L$ and an $L$-structure $\mathcal{A}$. Suppose $a, b \in dom(\mathcal{A})$. Then

$a$ is weakly discernible from $b$ in $\mathcal{A}$ iff there is an $L$-formula $\phi(x,y)$ such that $\mathcal{A} \models \forall x \neg \phi(x,x)$ and $\mathcal{A} \models \phi(a,b)$.

Saturday, 13 August 2011

Some metaphysicians have wanted to reduce "is" to something else (e.g., Saunders 2003, Ladyman 2005). By "is", I mean "is" as in

6 + 9 is 15
Sir Walter Scott is the author of Waverley.

I don't mean "is" as in

Frank Plumpton Ramsey is the Amy Winehouse of 20th century philosophy
Tuna is the chicken of the sea.

Mathematicians use the symbol "=" for "is", so the question is what is "="? (Maybe, better: what is =?). The most popular answer has usually been: "=" is indiscernibility. More exactly,

$x = y$ iff $x$ and $y$ have all the same properties.

One direction of this (right-to-left) is known as Leibniz's Principle of Identity of Indiscernibles. It can be given a formalization as follows:

(PII) $x = y \Leftrightarrow \forall X(Xx \rightarrow Xy)$

This definition of $=$ is fine with me, but it is a bit conceptually circular (see Savellos 1990 for a useful discussion). For its truth rests on treating being $x$ as a property of $x$. (Known as the haecceity of $x$.) The extension of the property of being $x$ is just the unit set, $\{x\}$. If one's property quantifier ranges over all of these properties (or, equivalently, unit sets), then the above definition comes out true. But if one examines the proof, one needs a comprehension instance using the = predicate. One can find counter-examples if one uses a Henkin model $(\mathcal{A}, S)$ in which = is not first-order definable in $\mathcal{A}$, and whose second-order domain $S$ lacks the required unit sets (see Ketland, 2006, "Structuralism and the Identity of Indiscernibles", Analysis 66.4, pp. 313-4). A simple example of such a structure $\mathcal{A}$ has $dom(\mathcal{A}) = \{0,1\}$ and a single binary relation $R = \{(0,0),(0,1), (1,0), (1,1)\}$. This example is discussed also in Button 2006.

In addition to the Leibnizian definition, there are others, invoking different notions of indiscernibility. Given a formalized language $L$ with just predicates, one can define, for each primitive predicate $P$, a notion of "being indiscernible relative to $P$". If there are finitely many different predicates in $L$, one can take the conjunction of these to give a formula $x \approx y$, which expresses the strongest notion of indiscernibility expressible in $L$. (This approach was first given in Hilbert & Bernays, Grundlagen der Mathematik (1934, Vol 1), and was later discussed on several occasions by Quine.) One can prove various things about this formula. Suppose we are given a language $L$ and $L$-structure $\mathcal{A}$. We can show that the formula $x \approx y$ behaves just like an identity predicate (it satisfies reflexivity and substitutivity); and if $=$ is definable at all in $\mathcal{A}$ by some formula $\phi(x,y)$, then $=$ is definable by $x \approx y$. One may also define two further notions of indiscernibility, which can be shown to be equivalent to $x \approx y$. One is

"$x$ is not weakly discernible from $y$"

(the original idea is due to Quine: see Quine 1976). The other is

"$x$ is polyadically indiscernible from $y$".

And then one can also give examples of structures $\mathcal{A}$ which contain indiscernible but distinct elements (i.e., $=$ is not first-order definable in $\mathcal{A}$). I call such structures non-Quinian and I call structures in which $=$ is definable Quinian. Since $x \approx y$ expresses an equivalence relation on $\mathcal{A}$, one can take the quotient of $\mathcal{A}$ under this relation. In the resulting structure, $=$ is now definable. In reverse, one can take a Quinian structure, and add "indiscernible elements", to get a structure with distinct but indiscernible elements. It's worth stressing that whether identity is definable or not in some interpreted language $L$ depends on the properties definable in $L$.

This is mathematical logic (for more technical results about defining identity and indiscerniblity notions, see Ketland 2006 and Ketland 2011, "Identity and Indiscernibility", Review of Symbolic Logic 4, and A. Caulton and J. Butterfield 2011, "On Kinds of Indiscernibility in Logic and Metaphysics", BJPS). But on the underlying philosophical issue, why should we have to define, or reduce, "=" anyway? Why not take "=" to be a primitive notion? Why not say,

Is is just what it is, and not another thing?

This is, more or less, Frege's suggestion in his 1891 review of Husserl and endorsed in Savellos 1990. I put it, in 2006, like this:

Must the identity relation on positions be defined in terms of the other distinguished relations? Or might the identity relation for positions be taken as primitive? For my part, I see no compelling reason why the identity relation, in general, should not be thought of as primitive. The reasons sometimes given for not taking identity as primitive seem to me to be anti-realist, reductionist or verificationist in spirit. The contrary view, that identity is primitive and indefinable, was advocated by Gottlob Frege, in his 1891 review of Edmund Husserl’s Philosophie der Arithmetik: ‘since any definition is an identity, identity itself cannot be defined’ (see Geach & Black 1980, p. 80). In an illuminating article, Elias Savellos has similarly argued that ‘identity must be viewed as an indefinable, primitive notion’ because ‘any attempt to define identity is bound to be circular, since the intelligible understanding of the notion of identity must make recourse to the intelligible understanding of identity itself’ (1990, p. 476). (Ketland 2006, pp. 305-6.)

In general, I'm not keen on reductionism. Bad for the health. That isn't to say we don't know how to reduce $A$s to $B$s (or, reduce theory $T$ in $L$ to theory $T^{\prime}$ in $L^{\prime}$, or, to interpret one structure $\mathcal{A}$ in another $\mathcal{B}$).

One can reduce natural numbers to finite sets (usually, reduce $\mathbb{N}$ to $\omega$.) One can reduce integers to equivalence classes of pairs of natural numbers. For example, identify $\mathbb{Z}$ with the set $\{[(n, m)]: n, m \in \mathbb{N}\}$, where $[(n, m)]$ is the equivalence class of pairs $(k, p)$ such that $n+p = k+m$. The intuitive idea is that the ordered pair $(n,m)$ represents the integer $n-m$. But there are infinitely many such; so we take the equivalence class of all. By a similar method (this time with $\times$ instead of $+$), we get the rationals, $\mathbb{Q}$. One can reduce real numbers to equivalence classes of Cauchy sequences of rationals, or to Dedekind cuts of rationals. One can reduce the ordered pair $(x,y)$ to some set, say $\{\{x\}, \{x, y\}\}$. One can reduce ordered pairs of numbers (and, more generally, finite sequences of numbers) to numbers. See what I did? - I conflated ordered $n$-tuples with finite sequences. No matter: we can reduce either to the other. And relations can be reduced to sets of ordered $n$-tuples and functions to special relations satisfying a uniqueness condition on one argument place. One can reduce ordinals to transitive $\epsilon$-well-founded sets and reduce cardinals to ordinals.

Even so, I don't think numbers are sets; I don't think real numbers are equivalence classes of Cauchy sequences; etc. These reductions are examples of interpretations or encodings - successful ones, but generally non-unique. This is the message of Paul Benacerraf's famous 1965 article, "What Numbers Could Not Be" (Philosophical Review 74). Given the non-uniqueness of these reductions, one can perhaps try and argue that one is special or distinguished, but this is quite implausible (despite the convenience of identifying $\mathbb{N}$ with $\omega$.) Or one can argue that there are no numbers, integers, ordered pairs, etc., and all there are are sets or classes (and so we consider mathematical theories to be convenient definitional extensions of standard set theory, while agreeing that these extensions are not unique). That seems to be Quine's view. Or one can think of these special mathematical entities as sui generis. Or one can think of them structurally (e.g., to be a real number is to be a node in the abstract real number structure). This is Shapiro's and Resnik's view. Or one can be a nominalist (there are no natural numbers, rationals, reals, sets, functions, etc.). This is Field's view.

(Constructivism, as an ontological claim (e.g., Heyting), seems indefensible: if numbers are "mental constructions", then either there are infinitely many mental constructions (one for $0$, one for $1$, and so on) or only finite many numbers. The former contradicts empirical fact and the second contradicts mathematical fact. Perhaps numbers are "possible" mental constructions, in the mind of God? Well, then one might as well just admit that they are not constructions after all. Constructivism, as an epistemological claim, concerning which proofs are legitimate, is not affected by this argument.)

The views are: sets are basic and there's a special reduction of mathematical entities to sets (implausible, I think); there are only sets and classes and all other mathematical entities are introduced by convenient but non-unique definitions (Quine); different kinds of mathematical entities are sui generis; mathematical entities are nodes/places/positions in abstract structures (Shapiro); mathematical entities are mental constructions (Heyting); there aren't any mathematical entities (Goodman, Field).

Of these, I think my preferred view is the sui generis one. Nominalism and constructivism are inconsistent with science. Quine's view is too austere. Structuralism is a bit too weird (the "nodes" in the domain of an abstract structure have an odd status, but maybe structuralism is sui generis-ism in disguise).

So, the natural numbers are just what they are, not another thing, which brings me to Bishop Butler:

If the observation be true, it follows, that self-love and benevolence, virtue and interest, are not to be opposed, but only to be distinguished from each other; in the same way as virtue and any other particular affection, love of arts, suppose, are to be distinguished. Every thing is what it is, and not another thing. (Bishop Joseph Butler 1726, Fifteen Sermons Preached at the Rolls Chapel. Preface.)

Ok, so that's not mathematics, I admit; but, as Tom Lehrer once joked, the idea's the important thing.